Fast and Perfect Sampling of Subgraphs and Polymer Systems

Optimal identifying codes of two families of Cayley graphs

Several applications in biology - e.g., incorporation of protein flexibility in ligand docking algorithms, interpretation of fuzzy X-ray crystallographic data, and homology modeling - require computing the internal parameters of a flexible fragment (usually, a loop) of a protein in order to connect its termini to the rest of the protein without causing any steric clash. One must often sample many such conformations in order to explore and adequately represent the conformational range of the studied loop. While sampling must be fast, it is made difficult by the fact that two conflicting constraints - kinematic closure and clash avoidance - must be satisfied concurrently. This paper describes two efficient and complementary sampling algorithms to explore the space of closed clash-free conformations of a flexible protein loop. The "seed sampling" algorithm samples broadly from this space, while the "deformation sampling" algorithm uses seed conformations as starting points to explore the conformation space around them at a finer grain. Computational results are presented for various loops ranging from 5 to 25 residues. More specific results also show that the combination of the sampling algorithms with a functional site prediction software (FEATURE) makes it possible to compute and recognize calcium-binding loop conformations. The sampling algorithms are implemented in a toolkit (LoopTK), which is available at https://simtk.org/home/looptk.

Shuffling algorithm for boxed plane partitions

Variable selection is very important in many fields, and for its resolution many procedures have been proposed and investigated. Among them are Bayesian methods that use Markov chain Monte-Carlo (MCMC) sampling algorithms. A problem with MCMC sampling, however, is that it cannot guarantee that the samples are exactly from the target distributions. This drawback is overcome by related methods known as perfect sampling algorithms. In this paper, we propose the use of two perfect sampling algorithms to perform variable selection within the Bayesian framework. They are the sandwiched coupling from the past (CFTP) algorithm and the Gibbs coupler. We focus our attention to scenarios where the model coefficients and noise variance are known. We indicate the condition under which the sandwiched CFTP can be applied. Most importantly, we design a detailed scheme to adapt the Gibbs coupler algorithm to variable selection. In addition, we discuss the possibilities of applying perfect sampling when the model coefficients and noise variance are unknown. Test results that show the performance of the algorithms are provided.

Counting 1-factors in infinite graphs

This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda$</tex> then spectral independence holds at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda$</tex> . As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n\ \text{log}\ n)$</tex> mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvi-nok (2017), refined by Patel and Regts (2017). There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O$</tex> ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> log n) sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O$</tex> ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> log n) sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.

By the method of nonstandard analysis, the definition of *-finite graph is given, and necessary and sufficient conditions of *-finite graph are obtained. Further, by the Transfer Principle, we apply the theory of finite graph to *-finite graph, embed given infinite graph into some *-finite graph, and finally obtain the related results of infinite graph.

Cycles through all finite vertex sets in infinite graphs

For d ≥ 2 and all q≥ q 0(d) we give an efficient algorithm to approximately sample from the q-state ferromagnetic Potts and random cluster models on the torus (ℤ / n ℤ ) d for any inverse temperature β≥ 0. This stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen–Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models.

On the Metric Dimension of Infinite Graphs

Let G be a graph and let s be a vertex of G. We consider the structure of the set of all lifts of two edges incident with s that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least $$(\deg (s)-1)/2$$(deg(s)-1)/2 disjoint pairs that lift. We consider the lifting graph: its vertices are the edges incident with s, two being adjacent if they form a liftable pair. We have three main results, the first two with the same hypotheses as for Mader's Theorem. (i) Let F be a subset of the edges incident with s. We show that F is independent in the lifting graph of G if and only if there is a single edge-cut C in G of size at most $$r+1$$r+1 containing all the edges in F, where r is the maximum number of edge-disjoint paths from a vertex (not s) in one component of $$G-C$$G-C to a vertex (not s) in another component of $$G-C$$G-C. (ii) In the k-lifting graph, two edges incident with s are adjacent if their lifting leaves the resulting graph with the property that any two vertices different from s are joined by k pairwise edge-disjoint paths. If both $$\deg (s)$$deg(s) and k are even, then the k-lifting graph is a connected complete multipartite graph. In all other cases, there are at most two components. If there are exactly two components, then each component is a complete multipartite graph. If $$\deg (s)$$deg(s) is odd and there are two components, then one component is a single vertex. (iii) Huck proved that if k is odd and G is $$(k+1)$$(k+1)-edge-connected, then G is weakly k-linked (that is, for any k pairs $$\{x_i,y_i\}$${xi,yi}, there are k edge-disjoint paths $$P_i$$Pi, with $$P_i$$Pi joining $$x_i$$xi and $$y_i$$yi). We use our results to extend a slight weakening of Huck's theorem to some infinite graphs: if k is odd, every $$(k+2)$$(k+2)-edge-connected, locally finite, 1-ended, infinite graph is weakly k-linked.

It is well known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalize the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large minimum degree at the vertices imply the existence of large complete (topological) minors in infinite graphs with countably many ends.

The tropospheric duct parameter inversion problem using the global optimization algorithm to find the optimal value of the objective function is a typical point estimation problem. In some cases, we need not only the best estimate of the refractive profile, but also the probability problem of the most advantageous, that is, the uncertainty of the inversion. Therefore, this paper uses Bayesian theory to calculate the uncertainty of the statistics of duct profile parameters. However, since the parameter vector dimension is high, the posterior probability density of each parameter cannot be directly calculated. Therefore, an efficient sampling algorithm is needed to sample the parameter vector. We used the Metropolis-Hasting sample in the MCMC sampling algorithm to sample the profile parameters and obtained a good statistical result.

The reconstruction of the structure of biological tissue using electromyographic (EMG) data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, EMG data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis–Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes’ theorem proves the well-posedness of the Bayesian inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis–Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.

In previous work, we described a Markovian state model (MSM) for analyzing molecular-dynamics trajectories, which involved grouping conformations into states and estimating the transition probabilities between states. In this paper, we analyze the errors in this model caused by finite sampling. We give different methods with various approximations to determine the precision of the reported mean first passage times. These approximations are validated on an 87 state toy Markovian system. In addition, we propose an efficient and practical sampling algorithm that uses these error calculations to build a MSM that has the same precision in mean first passage time values but requires an order of magnitude fewer samples. We also show how these methods can be scaled to large systems using sparse matrix methods.